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{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-22T08:40:03","documenter_version":"1.4.1"}}
{"documenter":{"julia_version":"1.10.4","generation_timestamp":"2024-06-28T17:25:57","documenter_version":"1.5.0"}}
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<!DOCTYPE html>
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Getting Started · RegularizedLeastSquares.jl</title><meta name="title" content="Getting Started · RegularizedLeastSquares.jl"/><meta property="og:title" content="Getting Started · RegularizedLeastSquares.jl"/><meta property="twitter:title" content="Getting Started · RegularizedLeastSquares.jl"/><meta name="description" content="Documentation for RegularizedLeastSquares.jl."/><meta property="og:description" content="Documentation for RegularizedLeastSquares.jl."/><meta property="twitter:description" content="Documentation for RegularizedLeastSquares.jl."/><meta property="og:url" content="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/gettingStarted/"/><meta property="twitter:url" content="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/gettingStarted/"/><link rel="canonical" href="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/gettingStarted/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">RegularizedLeastSquares.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li class="is-active"><a class="tocitem" href>Getting Started</a></li><li><a class="tocitem" href="../solvers/">Solvers</a></li><li><a class="tocitem" href="../regularization/">Regularization</a></li><li><span class="tocitem">API</span><ul><li><a class="tocitem" href="../API/solvers/">Solvers</a></li><li><a class="tocitem" href="../API/regularization/">Regularization Terms</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Getting Started</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Getting Started</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/blob/master/docs/src/gettingStarted.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Getting-Started"><a class="docs-heading-anchor" href="#Getting-Started">Getting Started</a><a id="Getting-Started-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-Started" title="Permalink"></a></h1><p>To get familiar with the different aspects of RegularizedLeastSquares.jl, we will go through a simple example from the field of Compressed Sensing.</p><p>In Addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, Images.jl and Random.jl, as well as PyPlot for visualization.</p><pre><code class="language-julia hljs">using RegularizedLeastSquares, LinearOperatorCollection, Images, PyPlot, Random</code></pre><p>To get started, let us generate a simple phantom</p><pre><code class="language-julia hljs">N = 256
<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Getting Started · RegularizedLeastSquares.jl</title><meta name="title" content="Getting Started · RegularizedLeastSquares.jl"/><meta property="og:title" content="Getting Started · RegularizedLeastSquares.jl"/><meta property="twitter:title" content="Getting Started · RegularizedLeastSquares.jl"/><meta name="description" content="Documentation for RegularizedLeastSquares.jl."/><meta property="og:description" content="Documentation for RegularizedLeastSquares.jl."/><meta property="twitter:description" content="Documentation for RegularizedLeastSquares.jl."/><meta property="og:url" content="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/gettingStarted/"/><meta property="twitter:url" content="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/gettingStarted/"/><link rel="canonical" href="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/gettingStarted/"/><script data-outdated-warner src="../assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/fontawesome.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/solid.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/6.4.2/css/brands.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css" rel="stylesheet" type="text/css"/><script>documenterBaseURL=".."</script><script src="https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.6/require.min.js" data-main="../assets/documenter.js"></script><script src="../search_index.js"></script><script src="../siteinfo.js"></script><script src="../../versions.js"></script><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-mocha.css" data-theme-name="catppuccin-mocha"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-macchiato.css" data-theme-name="catppuccin-macchiato"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-frappe.css" data-theme-name="catppuccin-frappe"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/catppuccin-latte.css" data-theme-name="catppuccin-latte"/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-dark.css" data-theme-name="documenter-dark" data-theme-primary-dark/><link class="docs-theme-link" rel="stylesheet" type="text/css" href="../assets/themes/documenter-light.css" data-theme-name="documenter-light" data-theme-primary/><script src="../assets/themeswap.js"></script></head><body><div id="documenter"><nav class="docs-sidebar"><div class="docs-package-name"><span class="docs-autofit"><a href="../">RegularizedLeastSquares.jl</a></span></div><button class="docs-search-query input is-rounded is-small is-clickable my-2 mx-auto py-1 px-2" id="documenter-search-query">Search docs (Ctrl + /)</button><ul class="docs-menu"><li><a class="tocitem" href="../">Home</a></li><li class="is-active"><a class="tocitem" href>Getting Started</a></li><li><a class="tocitem" href="../solvers/">Solvers</a></li><li><a class="tocitem" href="../regularization/">Regularization</a></li><li><span class="tocitem">API</span><ul><li><a class="tocitem" href="../API/solvers/">Solvers</a></li><li><a class="tocitem" href="../API/regularization/">Regularization Terms</a></li></ul></li></ul><div class="docs-version-selector field has-addons"><div class="control"><span class="docs-label button is-static is-size-7">Version</span></div><div class="docs-selector control is-expanded"><div class="select is-fullwidth is-size-7"><select id="documenter-version-selector"></select></div></div></div></nav><div class="docs-main"><header class="docs-navbar"><a class="docs-sidebar-button docs-navbar-link fa-solid fa-bars is-hidden-desktop" id="documenter-sidebar-button" href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li class="is-active"><a href>Getting Started</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href>Getting Started</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/JuliaImageRecon/RegularizedLeastSquares.jl/blob/master/docs/src/gettingStarted.md#" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Getting-Started"><a class="docs-heading-anchor" href="#Getting-Started">Getting Started</a><a id="Getting-Started-1"></a><a class="docs-heading-anchor-permalink" href="#Getting-Started" title="Permalink"></a></h1><p>To get familiar with the different aspects of RegularizedLeastSquares.jl, we will go through a simple example from the field of Compressed Sensing.</p><p>In Addtion to RegularizedLeastSquares.jl, we will need the packages LinearOperatorCollection.jl, Images.jl and Random.jl, as well as PyPlot for visualization.</p><pre><code class="language-julia hljs">using RegularizedLeastSquares, LinearOperatorCollection, Images, PyPlot, Random</code></pre><p>To get started, let us generate a simple phantom</p><pre><code class="language-julia hljs">N = 256
I = shepp_logan(N)</code></pre><p>In this example, we consider an operator which randomly samples half of the pixels in the image. Such an operator and the corresponding measurement can be generated by calling</p><pre><code class="language-julia hljs"># sampling operator
idx = sort( shuffle( collect(1:N^2) )[1:div(N^2,2)] )
A = SamplingOp(eltype(I), pattern = idx , shape = (N,N))
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y = A*vec(I)</code></pre><p>To recover the image, we solve the TV-regularized least squares problem</p><p class="math-container">\[\begin{equation}
\underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{y} \vert\vert_2^2 + λTV(\mathbf{x}) .
\end{equation}\]</p><p>For this purpose we build a TV regularizer with regularization parameter <span>$λ=0.01$</span></p><pre><code class="language-julia hljs">reg = TVRegularization(0.01; shape=(N,N))</code></pre><p>To solve the CS problem, the Alternating Direction Method of Multipliers can be used. Thus, we build the corresponding solver</p><pre><code class="language-julia hljs">solver = createLinearSolver(ADMM, A; reg=reg, ρ=0.1, iterations=20)</code></pre><p>and apply it to our measurement</p><pre><code class="language-julia hljs">Ireco = solve!(solver,y)
Ireco = reshape(Ireco,N,N)</code></pre><p>The original phantom and the reconstructed image are shown below</p><p><img src="../assets/sh.png" alt="Phantom"/> <img src="../assets/sh_tv.png" alt="Reconstruction"/></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../solvers/">Solvers »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.4.1 on <span class="colophon-date" title="Saturday 22 June 2024 08:40">Saturday 22 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
Ireco = reshape(Ireco,N,N)</code></pre><p>The original phantom and the reconstructed image are shown below</p><p><img src="../assets/sh.png" alt="Phantom"/> <img src="../assets/sh_tv.png" alt="Reconstruction"/></p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../">« Home</a><a class="docs-footer-nextpage" href="../solvers/">Solvers »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.5.0 on <span class="colophon-date" title="Friday 28 June 2024 17:25">Friday 28 June 2024</span>. Using Julia version 1.10.4.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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